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Inspiralling collisions

For the collision of inspiralling black holes there are no currently available full numerical simulations. Here the only information available is from the close limit approximation. In this case, one only expects to get correctly one portion of the waveform, since in addition to the final ringdown captured by the close limit approximation, one also will have the ``chirp'' and ``merger'' phases of the collision. Nevertheless, it is instructive to see what the use of the ``realistic'' ringdown part of the waveforms yields. The close-limit approximation is quite limited in validity for inspiralling black hole. The problem is that for large values of the angular momentum, as are expected in realistic collisions, the spacetime departs quite radically from that of a single spinning black holes. Best educated guesses suggest that trustworthy results from the close limit approximation can only be obtained up to values of the Kerr parameter of $a=0.5\,M$. At such values, for separations of a few $M$, the radiated energy is of the order of $0.3\%$ of the total mass of the system. Of the angular momentum, a similar fraction gets radiated away. By eyeballing the curve showing the energy dependence as a function of angular momentum, one could expect that a ``realistic'' collision with values of $a/M$ close to unity would probably radiate of the order of $1\%$ of the total mass. This is a significant extrapolation from the ``reliable'' results, but barring unexpected physics at the last moments of the collision, it is probably right. We present here the analysis of a ``close-limit'' type waveform for the ringdown of the final moments of an inspiralling collision. The waveform was calculated for $a=0.35\,M$ and separation $L=0.9\,M$ ($\mu_0=1.5$). The waveform is shown in figure [*].

Figure: The waveform for the ringdown of an inspiralling collision with Kerr parameter $a=0.35\,M$ and initial separation $L=0.9\,M$. Extrapolations beyond the realm of validity of the close limit suggest that for a ``realistic'' collision with $a/M$ close to unity about $1\%$ of the mass is radiated.
\begin{figure}\begin{center}
\epsfig{file=Figures/ring-close-insp.eps,height=2in}\end{center}\end{figure}

For the LIGO-I noise curve and a total black hole mass of $200\,M_\odot$, the fitting factor achieved by a quasinormal ringdown template with the ``correct'' $a=0.35\,M$ is $69\%$ (see figure [*]). However, this is not a true fitting factor in the sense that the maximization has been over the time of arrival only--for different values of the ringdown template mass and spin, the fit will improve. In particular, only the $\ell=m=2$ quasinormal mode is considered in constructing the ringdown template while the close-limit waveform contains excitations from the $\ell=2$, $m=0$ as well. For a rapidly spinning black hole, the $\ell=m=2$ mode will (eventually) dominate the waveform because it is much longer lived. However, the spin of the black hole in the present case is not particularly large, and the $m=0$ mode tends to dominate. In fact, the fitting factor obtained by using a ringdown template with $a=0$ is $84\%$--much better than the fit obtained using the $\ell=m=2$ ringdown with $a=0.35\,M$. It would be useful to examine the entire parameter space of quasinormal ringdowns in mass and spin as well as arrival time in order to obtain the ``true'' fitting factor.

Figure: Pure ringdowns superimposed on the waveform for the ringdown of an inspiralling collision with Kerr parameter $a=0.35\,M$ and initial separation $L=0.9\,M$ for the times of greatest correlation.
\begin{figure}\begin{center}
\index{colorpage}
\epsfig{file=Figures/ring-fits-insp.eps,height=2in}\end{center}\end{figure}

Authors: Jolien Creighton (jolien@tapir.caltech.edu) and Jorge Pullin (pullin@phys.psu.edu)


next up previous contents
Next: Example: ring-corr program Up: GRASP Routines: Black hole Previous: The close-limit approximation and   Contents
Bruce Allen 2000-11-19